Vol 8, No 10 (2017)

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 8(10), Oct. – 2017 199

SOME NEW FAMILIES OF ODD GRACEFUL GRAPHS

AMIYA KUMAR BEHERA*

1

_{, DEBDAS MISHRA}

2

_{AND PURNA CHANDRA NAYAK}

3

1

_{Department of Mathematics,}

Siksha ‘O’ Anusandhan University, Bhubaneswar, India.

2

_{Department of Mathematics,}

C. V. Raman College of Engineering, Bhubaneswar, India.

3

_{Department of Mathematics,}

Bhadrak (Autonomous) College, Bhadrak, India.

(Received On: 19-09-17; Revised & Accepted On: 16-10-17)

ABSTRACT

I

n this paper we give odd graceful labeling to Bistar, _:2

, 1n

K , K1,_n,_n , Coconut tree CT n,3 , K1,n+Kn,1 ,

n n

nC

K

K

,1

+

4

+

1, for

n

≥

3

, nC K1,n 2

4 + ,

K

1,n

+

nP

2

+

K

n,1 ,

K

1,n

+

nP

3

+

K

n,1 , C2r with every alternate

vertex attached to a pendant vertex .

Key Words: Odd graceful labeling, Vertex labeling, edge labeling.

1. INTRODUCTION

A graph G with q edgesis said to be odd-graceful, if there is an injection f from V(G) to

{

0,1,2,,2q−1

}

such that, when each edge {x, y} of G is assigned the label

f

( ) ( )

x

−

f

y

, the resulting edge labels are

{

1,3,5,,2q−1

}

. The concept of odd graceful graph was introduced in 1991 by Gnanajothi [1].

Gnanajothi [1] proved that the class of odd-graceful graphs lies between the class of interlaced graphs and the class of bipartite graphs by showing that every interlaced graph has an odd-graceful labeling and every graph with an odd cycle

is not odd-graceful. She also proved the following graphs are odd-graceful:

P

_n;

C

_n if and only if n is even;

K

_m,n;

combs 1 n

P K (graphs obtained by joining a single pendent edge to each vertex of

P

_n ); books ; crowns

1 n C K (graphs obtained by joining a single pendent edge to each vertex of

C

_n ) if and only if n is even ; the disjoint union of

copies of

C

₄; the one-point union of copies of

C

₄;

C

_n

×

K

₂if and only if n is even; caterpillars; rooted trees of height 2; the graphs obtained from

P

_n (n > 3) by adding exactly two leaves at each vertex of degree 2 of

P

_n; the

graphs obtained from

P

_n

×

P

₂ by deleting an edge that joins to end points of the

P

_n paths ; the graphs obtained from a star by adjoining to each end vertex the path

P

₃ or by adjoining to each end vertex the path

P

₄ . She conjectures that all trees are odd-graceful and proves the conjecture for all trees with order up to 10. Barrientos [2] has extended this to trees of order up to 12. For details in the progress made so far in this area one can refer to the latest Survey on graph labeling problems due to Gallian [3]. Here we are inspired by the works of Solairaju [4] and Moussa and Badr [5] and give odd graceful labeling to the graphs : Bistar, K_1,n:2 ,

n n

K1, , , Coconut tree CT n,3 , K1,n+Kn,1 ,

n

n nC K

K ,1+ 4+ 1, for

n

≥

3

, nC K1,n 2

4 + , K1,n +nP2+Kn,1 , K1,n +nP3+Kn,1 , C2r with every alternate vertex attached to a pendant vertex .

Corresponding Author: Amiya Kumar Behera*

1

_,

RESULTS

Theorem 2.1: A Bistar

B

_m,n is odd graceful.

Proof: Let

B

_m,_n be a Bistar containing

m

+

n

+

2

vertices consisting of the set

{

u u

1

,

2

,

v w

i

,

j

}

,

1,

,

i

=



m

,

j

=

1

,



,

n

where

u

₁ ,

u

₂ are the internal vertices and

v

_i,

w

_j are the vertices adjacent to

u

₁ ,

u

₂

respectively. Let q be the total number of edges of

B

_m,n such that q=m+n+1. Now the vertex labelling of the

Bistar

B

_m,n are given as

( )

u₁ =0

f ,f u

( )

₂ =1,

f v

( )

i

=

2

q

− +

2

i

1,

i

=

1

,

2

,



,

m

,

f

( )

w

j

=

2

q

−

2

m

−

2

j

+

2

,

j

=

1

,

2

,



,

n

.

The edge labelling of the Bistar is given as

f

( )

u

₁

v

_i

=

2

q

−

2

i

+

1

,

i

=

1

,

2

,



,

m

,f

(

u2wj

)

=2q−2m−2j+1,

j=1,2 ,…,n. So the edge labelling of the Bistar consists of the labelled set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence the Bistar

B

_m,_n is odd graceful.

Figure-1: A Bistar

B

_4,4 with an odd graceful labeling

Theorem 2.2: K_1,n:2 is odd graceful for every natural number n.

Proof: Let the set of vertices of K_1,n:2 be

{

u

,

v

,

w

,

u

_i

,

v

_i

}

, i=1,,n. Let P3 be the path with vertices {u , v, w},

where u, v, w are the internal vertices and

u

_i,

v

_j the vertices adjacent to u, v and w respectively. Let

(

K

_1,n

:

2

)

=

{

uw

,

vw

,

uu

_i

,

vv

_i

,

i

=

1

,

2

,



,

n

}

be then edge set of _:2

, 1n

K and q be the total number of edges

of _:2 , 1n

K such that

q

=

2

n

+

2

. The vertex labeling _:2 , 1n

K is given as follows

f

( )

u

i

=

2

i

−

1

,

i

=

1

,



,

n

,

( )

v

n

i

i

n

f

_i

=

2

(

+

)

+

1

,

=

1

,

2

,



,

, f

( )

u =0,f(v)=2,f(w)=4n+3, Now the induced edge labelling of _:2 , 1n

K

is as follows f

( )

uui =2i−1,i=1,,n ,

f

(

vv

j

)

=

2

(

n

+

j

)

−

1

,

j

=

1

,



,

n

.

f uw

(

)

=

4

n

+ =

3

2

q

−

1

( )

wv

=

4

n

+

1

=

2

q

−

3

f

. So the edge labelling of

K

_1,n

:

2

consists of the set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence

2

:

, 1n

K

is odd graceful.

© 2017, IJMA. All Rights Reserved 201 Theorem-2.3:

K

1,_n,_n is odd graceful for every natural number n.

Proof: Let the set of vertices of

K

1,_n,_n be

{

u

,

u

i

,

v

i

}

such that the root vertex u is adjacent to

u

i and

u

i is

adjacent to

v

_i for

i

=

1

,



,

n

. Let

E

(

K

_1,n,n

)

=

{

uu

_i

,

u

_i

v

_i

}

be the edge set of

K

1,n,n . The vertex labelling of

n n

K

_{1, ,} are given as

f

( )

u

=

0

,

f(u_i)=4n−2i+1,

f

(

v

_i

)

=

2

n

+

2

i

−

2

,

i

=

1

,



,

n

. Now the edge labelling

of

K

1,_n,_n are given as

f

( )

uu

i

=

4

n

−

2

i

+

1

=

2

q

−

2

i

+

1

,

f

(

u

i

v

j

)

=

2

n

−

4

i

+

3

,

i

=

1

,



,

n

. So the edge

labelling of

K

1,_n,_n consists of the set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence

K

1,n,n is odd graceful.

Figure-3:The tree with an odd graceful labeling

Theorem-2.4: The Coconut tree CT n,3 is odd graceful.

Proof: Let the set of vertices of

CT

n

,

3

be

{

u,v,w,ui,i=1,,n

}

where u, v, w lie on the central path of

3

,

n

CT

. Let

E

CT

n

,

3

=

{

uu

_i

,

uv

,

vw

}

,

i

=

1

,



,

n

be the edge set of

CT

n

,

3

. The vertex labelling

of

CT

n

,

3

are given as

f

( )

u

=

0

,

f

(

v

)

=

2

n

+

3

,

f

(

w

)

=

2

, ( )f ui = −2i 1

,

i

=

1

,



,

n

. Now the edge

labelling of

CT

n

,

3

are given as

f uv

( )

=

2

n

+

3

=

2

q

−

1,

f vw

(

)

=

2

n

+

1

=

2

q

−

3,

f uu

(

_i

)

= −

2

i

1,

1,

,

i

=



n

. So the edge labelling of

CT

n

,

3

consists of the set {1, 3, 5, - - - , 2q-1}. Hence the Coconut tree CT n,3 is odd graceful.

Figure-4: The coconut tree

CT

5

,

3

with an odd graceful labeling

Theorem-2.5: K_1,n+K_n,1 is odd graceful for every natural number n.

Now the vertex labelling of

1 , , 1n Kn

K + is given by

f u

( )

=

0, ( )

f v

=

2 ,

n

f u

( )

i

=

4

n

− +

2

i

1,

i

=

1,



,

n

.

And the edge labelling of

K

_1,n

+

K

_n,1 is given by

f uu

( )

_i

=

4

n

− + =

2

i

1, 2

q

− +

2

i

1

f u v

(

_i

)

=

2

n

− +

2

i

1,

n

i

=

1

,



,

. So the edge labelling of

K

_1,n

+

K

_n,1 consists of the set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence

1 , , 1n Kn

K +

is odd graceful.

Figure-5: The graph

1 , 6 6 , 1 K

K + with an odd graceful

Theorem-2.6:

K

_n,1

+

nC

4

+

K

1,_n is odd graceful for every natural number n ≥ 3.

Proof: Let the set of vertices of

n n nC K

K _,1+ ₄+ _1, be

{

u

,

v

,

u

_i

,

v

_i

,

w

_i

,

i

=

1

,



,

n

}

where the vertices

are in the order

u

_i

−

u

−

v

_i

−

v

−

w

_i . Then edge set of

K

_n,1

+

nC

₄

+

K

_1,n is

{

u u uv v v vw

_i

,

_i

,

_i

,

_i

}

,

1,

,

i

=



n

. We define the vertex labelling f of G by f

( )

u =0,

f

( )

v

=

8

n

−

2

,

( )







=

+

−

=

=

n

i

i

n

i

u

f

i

,

,

2

,

3

2

8

1

,

1



( )

v

n

i

i

n

f

_i

=

12

−

2

+

1

,

=

1

,



,

2

( )

i 2 2 3 , 0 , , 1.

f w = n+ i− i= − − − −n

The label of the edges of

K

_n,1

+

nC

₄

+

K

_1,n are given by

( )

1

,

1

,

8

2

3 ,

2, 3,

,

i

i

f u u

n

i

i

n

=



= 

_{− +}

₌

_{− − −}



( )

vw

=

6

n

−

2

i

+

1

,

i

=

0

,

−

−

−

,

n

−

1

f

i

( )

v

v

n

i

i

n

f

i

=

4

−

2

+

3

,

=

1

,



,

2

( )

_i

12

2

1,

1,

, 2 .

f uv

=

n

− +

i

i

= − − −

n

We observe that the label of the edges of

K

_n,1

+

nC

₄

+

K

_1,n constitute the set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence

n

n nC K

© 2017, IJMA. All Rights Reserved 203 Figure-6: The graph

K

_n,1

+

nC

₄

+

K

_1,n with an odd graceful

Theorem 2.7:

n K nC 1,

2

4 + is odd graceful.

Proof: Let the vertices of

n K nC 1,

2

4 + consists of the set

{

u

,

v

,

u

i

,

v

i

,

w

,

w

i

,

i

=

1

,



,

n

}

, where the vertices are adjacent in the order

u

−

u

_i

−

v

−

v

_i

−

w

−

w

_i. The edge set of

n

K

nC

₄2

+

_1, is

{

uu

_i

,

u

_i

v

,

vv

_i

,

v

_i

w

,

ww

_i

}

,

i

=

1

,



,

n

. The vertex labelling f of

_nC

₄2

+

_K

_1,n are assigned as

( )

u

4

n

,

f

=

f

(

u

_i

)

=

8

n

+

2

i

−

1

,

f

( )

v

=

0

,

f

( )

v

_i

=

20

n

−

2

i

+

1

,

i

=

1

,



,

2

n

,

f

( )

w

=

16

n

,

f

( )

w

_i

=

2

i

−

1

,

n

i

=

1

,



,

2

.The label of the edges of

nC

₄2

+

K

_1,n are given by

( )

uu =4n+2i−1,

f _i

f

(

u

_i

v

)

=

8

n

+

2

i

−

1

,

f

( )

vv

i

=

20

n

−

2

i

+

1

,

f

(

v

i

w

)

=

4

n

−

2

i

+

1

,

i

=

1

,



,

n

.

(

_i

)

16

2

1

f ww

=

n

− +

i

,i=1,...2n. We observe that the label of the edges of

nC

₄2

+

K

_1,n

constitute the

set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence

nC

K

1,_n 2

4

+

is odd graceful.

Theorem 2.8:

K

_1,n

+

nP

₂

+

K

_n,1 is odd graceful.

Proof: Let the vertices of K_1,n+nP₂+K_n,1 consists of the set

{

u

,

u

_i

,

v

,

v

_i

}

where the vertices are adjacent

in the order

u u

− − −

_i

v

_i

v

. The edge set of

K

_1,n

+

nP

₂

+

K

_n,1 is

{

u

u

_i

,

u

_i

v

_i

,

v

_i

v

;

i

=

1

,



,

n

}

. The labeling of the vertices of

1 , 2 ,

1n

nP

K

n

K

+

+

are given by

f

(

u

)

=

0

,f

( )

u_i =6n−2i+1

,

f

(

v

)

=

2

n

−

1

n i

i n v

f( _i) 4 4 2, 1, ,

, = − + =  . The labeling of the edges of

K

_1,n

+

nP

₂

+

K

_n,1 are given by

( )

uu

n

i

f

u

v

n

i

f

v

v

n

i

i

n

f

_i

=

6

+

2

+

1

,

(

_{i i}

)

=

2

+

2

−

1

,

(

_i

)

=

2

−

4

+

3

,

=

1

,



,

. Now the edge labeling of

1 , 2 ,

1n

nP

K

n

K

+

+

constitute the set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence the graph

K

_1,n

+

nP

₂

+

K

_n,1 is an odd graceful graph.

Figure-8: The graph

K

_1,4

+

4

P

₂

+

K

_4,1 with an odd graceful

Theorem 2.9:

K

_1,n

+

nP

₃

+

K

_n,1 is odd graceful.

Proof: Let the vertices of

1 , 3 ,

1n nP Kn

K + + consists of the set

{ , , ,

u u v w w

i i i

, }

where the vertices are adjacent in the order

u

−

u

_i

−

v

_i

−

w

_i

−

w

. The edge set of

1 , 3 ,

1n nP Kn

K + + is

{

uu u v v v

_i

,

_{i i}

,

_i

;

.

i

=

1,



, }

n

The

labeling of the vertices of

1 , 3 ,

1n

np

K

n

K

+

+

are given by

f u

( )

=

0 ,

f u

( )

i

=

8

n

− +

2

i

1,

f v

( )

i

=

4

n

− +

4

i

2,

( )

(

_i

)

6

2

1),

4

f w

=

n

− +

i

f w

=

n

. The labeling of the edges of K_1,n+nP₃+K_n,1 are given by

( )

_i 8 2 1, ( _{i i}) 4 2 1, ( _{i i}) 2 2 1,

f u u n= + i− f u v = n+ i− f v w = n+ i−

f w w

(

_i

)

=

2

n

− + =

2

i

1

i

1,



,

n

. We

observe that the labels of the edges of

K

_1,n

+

nP

₃

+

K

_n,1 constitute the set

{

1

,

3

,

5

,



,

2

q

−

1

}

. Hence the

graph

K

_1,n

+

nP

₃

+

K

_n,1 is an odd graceful graph.

Figure-9: The graph

K

_1,4

+

4

P

₃

+

K

_4,1with an odd graceful labeling

Theorem-2.10: The graph C2r with every alternate vertex attached to a pendant vertex is odd graceful.

© 2017, IJMA. All Rights Reserved 205

2

1, 3,..., 2

1

( )

2, 4, 6,..., 2

2

4

2

2

i

n i

if

i

r

f u

i

if

i

r

r

if

i

r

−

=

−





=

_

=

−

 −

=



and

0

1

( )

2

2

2

3

2

2

5, 7,..., 2

1

i

if

i

f u

n

i

if

i

n

i

if

i

r

=





=

_

− +

=



₋

₌

₋



( ) ( )

u

f

u

n

( )

i

i

f

_i−1

−

_i

=

2

−

−

1

−

=

2

n

−

2

i

+

1

,

i

=

2

,

3

,



,

2

r

−

1

;

f

(

u

_2r−1

) ( )

−

f

u

_2r

=

3

;

( ) ( )

u

₂

−

f

u

₁

=

2

n

−

1

−

(

4

r

−

2

)

f

_r =

6

r

−

1

−

(

4

r

−

2

)

=

2

r

+

1

,

f

( ) ( )

u

₁

−

f

v

₁

=

2

n

−

1

,

( ) ( )

u

₃

−

f

v

₃

=

1

f

,

f

(

u

_2i−1

) (

−

f

v

_2i−1

)

=

2

i

−

1

,

i

=

3

,

4

,



,

r

. Observe that the set of all vertex labels of

the graph is the set

{

1

,

3

,

5

,



,

2

n

−

1

}

. Hence, the given graph is odd graceful.

Figure 10:

C

₁₄ with every alternate vertices attached to a pendant vertex is odd graceful.

REFERENCES

1. Gnanajothi R.B., (1991), Topics in Graph Theory, Ph. D. Thesis, Madurai Kamaraj University, India. 2. Barrientos C., Odd Graceful Graphs, Preprint.

3. Gallian J.A., A dynamic survey of graph labeling, Electronic Journal of Combinatorics, DS6, Eighteenth edition, December 17, 2015, url:http://www.combinatorics.org/Surveys/.

4. Solaraju A., Edge – Odd Graceful Graphs, Electronic Notes in Discrete Mathematics, 33 (2009), PP. 15 – 20. 5. Solaraju A., Edge – Odd Graceful Labeling of Crown Graphs, Proceedings of 1st Int Conference on Computer

Science from Algorithms to Applications (CSAA-2009), 8-10 2009, Cairo, PP. 15 – 20.

6. Graf A, A new graceful labeling for pendant graphs, Aequationes Mathematicae DOI

10.1007/s00010-012-0184-4

Source of support: Nil, Conflict of interest: None Declared.